Acoustic impedance of micro-perforated panels in a grazing flowXiaoqi Zhang, and Li Cheng
Citation: The Journal of the Acoustical Society of America 145, 2461 (2019); doi: 10.1121/1.5098785View online: https://doi.org/10.1121/1.5098785View Table of Contents: https://asa.scitation.org/toc/jas/145/4Published by the Acoustical Society of America
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Acoustic impedance of micro-perforated panels in a grazing flow
Xiaoqi Zhang and Li Chenga)
Department of Mechanical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China
(Received 3 January 2019; revised 2 April 2019; accepted 2 April 2019; published online 29 April2019)
Micro-perforated panels (MPPs) are widely used for broadband sound absorptions. For a MPP
exposed to a grazing flow, existing acoustic impedance formulas based on different flow parame-
ters give inconsistent results, thus calling for a systematic investigation of the issue to find more
intrinsic flow parameters allowing for a reliable acoustic impedance prediction. In this study,
three-dimensional CFD simulations are conducted on a MPP hole with a backing space in a flow
duct. Numerical results allow identifying the flow velocity gradient in the viscous sublayer as
the intrinsic flow parameter and show its linear relationship with a flow-related term in the
acoustic resistance formula. Through a linear regression analysis, an acoustic resistance formula
is established within a certain flow range (Mach number up to 0.25) under the linear acoustic
regime. The validity of the impedance formula is demonstrated through comparisons with
existing results and experimental data reported in the literature, showing good agreement and
superiority in terms of the prediction accuracy. VC 2019 Acoustical Society of America.
https://doi.org/10.1121/1.5098785
[NX] Pages: 2461–2469
I. INTRODUCTION
A micro-perforated panel (MPP) is a thin sheet with per-
forations over its surface. With the hole size typically in the
sub-millimeter range, a MPP provides a high acoustic resis-
tance (loss) and a low acoustic reactance (mass) by the struc-
ture itself, conducive to effective sound absorption with a
proper design. Owing to their unique and environmentally
friendly nature, MPPs are being widely used in various noise
control applications as an alternative to traditional porous/
fibrous sound absorption materials.
The acoustic property of a MPP can be characterized
by its surface impedance. Based on the work of Lord
Rayleigh1 and Crandall,2 Maa proposed various forms of
acoustic impedance prediction formulas by considering a
MPP as a lattice of short tubes with end corrections.3,4
Since then, research on MPPs has been intensifying, as evi-
denced by the large amount of papers published on the
topics, ranging from the design of various types of MPP
absorbers to the exploration of their applications.5,6 In addi-
tion to the conventional architectural and environmental
acoustic problems,7 their applications in compact mechani-
cal systems8,9 start to draw increasing attention. Affected
by the surrounding acoustic environment, MPPs behave
very differently from the simple laboratory setting, thus
requiring their consideration as an integral part of the entire
acoustic/vibro-acoustic system.10,11
In addition to problems in a still acoustic medium, an
important category of MPP problems involves flow, among
which the case of the grazing flow is probably the most rep-
resentative and practically important. Typical examples
include vehicle exhaust silencers, flow duct linings, or vari-
ous types of domestic products.
Existing work relating to the flow passing through a per-
forated hole in a plate has been arousing wide interest.
Theoretical12–16 or semi-theoretical models17,18 were devel-
oped upon making different simplifications on the interac-
tions between the acoustic waves and the flow field near the
perforated holes. While shedding light on the underlying
physics, most of these theoretical or semi-theoretical models
only consider inviscid flow for a hole size which is beyond
micro-perforated range. Meanwhile, due to the problem sim-
plification, the predicted acoustic impedance only qualita-
tively agrees with experimental data.19 A continuous effort
is to develop empirical models for the acoustic impedance
prediction.19–30 To this end, identifying the key flow parame-
ters which are intrinsically linked to the acoustic impedance
of the hole is the critical component. Observing the inconsis-
tencies in the prediction results,19 studies point at the neces-
sity of considering boundary layer parameters to
characterize the grazing flow effects. When the boundary
layer is thicker than the orifice diameter, the inner boundary
layer parameters need to be included as suggested by
Cummings.26
Most of the aforementioned studies considered perfora-
tions with a hole diameter typically around 1 mm or larger.
This, in a strict sense, falls beyond the scope of the micro-
perforation in the perspective of achieving sound absorption.
More relevant to the micro-perforation investigated in the
present paper is the work of Allam and Abom24 in which an
impedance formula was developed using the Mach number
as the flow parameter.
With the fast development of the computational capabil-
ity, numerical methods such as DNS, LES, and RANS are
more frequently used to investigate these problems.31–37 For
example, DNS was applied to a single Helmholtz resonator
to understand its sound energy dissipation mechanism.31–33
Results indicated a transition from the viscous-dominanta)Electronic mail: [email protected]
J. Acoust. Soc. Am. 145 (4), April 2019 VC 2019 Acoustical Society of America 24610001-4966/2019/145(4)/2461/9/$30.00
dissipation in the shear layer near the hole to the chaotic vor-
tex shedding dominant dissipation with the increasing sound
pressure level (SPL). It was shown that,35–37 without using
the highly computationally costly methods like LES or DNS,
more cost-effective tools like three-dimensional (3D)
URANS CFD could reasonably predict the acoustic behavior
of the Helmholtz resonator35 and the perforated panel
silencers36 in both no-flow and flow conditions.
Existing work allowed a qualitative description of the
acoustic behavior of MPPs in the presence of a grazing flow.
Different from the no-flow condition, however, a universally
accepted and consistent acoustic impedance formula to guide
the practical design of MPPs in the presence of flow is still
lacking. The intrinsic flow parameters which can intrinsi-
cally characterize the inherent grazing flow effects on the
acoustic impedance of MPPs still need to be found.
Motivated by this, the present paper investigates the
acoustic behavior of MPPs under fully developed turbulent
grazing flow conditions within the linear acoustic regime.
The main focuses are put on three aspects. (1) The flow field
around the holes of a MPP is scrutinized and investigated.
(2) A new flow parameter which intrinsically relates the
grazing flow effects with the acoustic impedance of the MPP
is identified. (3) A new impedance prediction formula is pro-
posed to supplement the one by Maa in the no-flow
condition.4
More specifically, numerical studies are carried out
through 3D URANS CFD simulations. Computed results are
then presented for validating the CFD model, identifying the
new intrinsic flow parameter and establishing the impedance
formula. Prediction results from the proposed formula are
then compared with those from other existing formulas or
experimentally measured data to show the accuracy and the
improvement that the new impedance formula brings about.
II. COMPUTATIONAL MODEL AND SIMULATIONMETHOD
A. Model
Consider a MPP panel with a honeycomb backing cav-
ity. Each hole in the face plate and the cavity cell behind it
forms a Helmholtz resonator (Fig. 1). The entire honeycomb
MPP absorber can be regarded as an assembly of an array of
Helmholtz resonators. We assume a low perforation ratio
(typically around 1%) so that each resonator would act inde-
pendently and the interaction between the holes is therefore
neglected. Under this condition, the acoustic impedance of
the panel is equal to that of a single isolated MPP hole
divided by the perforation ratio. The computational model
will use a single Helmholtz resonator, shown in Fig. 1.
Geometrical parameters of MPPs being investigated
hereafter are listed in Table I. All panels used for simulations
and comparisons, both numerically and experimentally, are
typical MPPs (with hole diameters smaller than 1 mm). Note
that Panel 1, with cylindrical holes of a diameter
d¼ 0.68 mm, is the configuration used by Malmary et al.38
All panels have the same size of 24� 24 mm, with a 10 mm
deep backing cavity, same as Ref. 38. The side length of
each MPP cell, L, is determined based on the perforation
ratio, giving L¼ 5.12 mm.
The modelled system is shown in Fig. 2 (two-dimen-
sional view). A MPP cell is flush-mounted on one sidewall
of a square duct with a cross section of 24� 24 mm. The
upstream portion of the duct is 1000 mm long, which allows
the flow to be fully developed. The downstream portion con-
tains two acoustic wavelengths as suggested by Ref. 39.
The computational domain is discretized by multi-size
grids, by using structured and unstructured meshes for the
duct and the backing cavity, respectively. Denser meshes are
used in the region near the orifice and the wall of the duct.
Away from these regions, the mesh size increases gradually.
To ensure an accurate description of the flow field near the
wall, the mesh size of the first layer adjacent to the duct wall
was designed to ensure that yþ ¼ yUs=v � 1, in which Us is
the friction velocity, y is the distance from the first layer
mesh to the wall in the normal wall direction, and v is the
kinematic viscosity of the air.
Inside the hole, the average size of the elements is
0.015 mm. In the duct, the grid spacing in the stream-wise
direction ranges from 0.015 to 1.6 mm upstream the resona-
tor and from 0.015 to 2.4 mm downstream, which results in a
minimum of about 65 and 43 grids per acoustic wavelength
upstream and downstream the resonator, respectively, at the
highest frequency considered in the study. The grid spacing
in the normal wall and span-wise direction ranges from
0.0075 to 1 mm and from 0.015 to 0.8 mm, respectively.
This results in a total of about 3 000 000 elements. The con-
vergence of the solution in relation to meshing is conducted
through investigating the time-domain variation of acoustic
pressure and velocity at the hole inlet section, which are
used to calculate the acoustic impedance of the hole. Results
indicate that convergence can be achieved to the accuracy
needed for impedance prediction.
B. Computational method
The 3D URANS is adopted using the commercial CFD
code, FLUENT. The choice of the simulation tool is justified
FIG. 1. (Color online) Perforated panel (left) modeled by a single
Helmholtz resonator (right).
TABLE I. Geometry parameters of the MPPs, Plate 1 is taken from Ref. 38.
Panel
number
Orifice diameter
d (mm)
Panel thickness
t (mm) t/d
Perforated
ratio d
1 0.68 1.02 1.7 1.39%
2 0.5 1.02 2.04 1.39%
3 0.3 0.3 1 1.39%
2462 J. Acoust. Soc. Am. 145 (4), April 2019 Xiaoqi Zhang and Li Cheng
by the observations made in previous work on acoustic
attenuators.35–37 The mass flow rate boundary condition is
applied to the inlet of the duct. The pressure at the outlet is
set to be the atmospheric pressure. Solid wall boundary con-
ditions with no slip are imposed to all wall surfaces. A
pressure-based implicit solver is employed. The pressure-
velocity coupling scheme Pressure-Implicit with Splitting of
Operators algorithm is applied for both the stable mean flow
field and the aero-acoustic coupling simulation. The second
order scheme is chosen for both the spatial discretization and
time integration. The number of unknowns in the equations
of URANS is more than that of the equations. These equa-
tions are unclosed. The Realizable k - e turbulence model is
adopted for this closure problem.
A two-step approach is used to deal with the interaction
between the flow and the acoustic wave near the hole. The sta-
ble mean flow field is first computed. Harmonic acoustic waves
with a given velocity amplitude are then added as the acoustic
excitation. The stable mean grazing flow is calculated by run-
ning the unsteady solver until flow properties stop changing.
The time step size for the stable mean flow computation is
5� 10�6 s. That used in acoustic simulations is 5� 10�7 s.
III. CFD MODEL VALIDATION
The viscous sub-layer region adjacent to the flow field
in the hole plays a key role in the acoustic impedance of the
hole. The velocity gradient in this layer and the friction
velocity are important parameters in this near-wall region.
They are first investigated to ensure a truthful description by
the CFD model.
A. Mean stream-wise velocity
To check the quality of the flow simulation, the mean
stream-wise velocity profiles at two upstream locations at
M¼ 0.25 are shown in Fig. 3. The origin of coordinates was
set at the center of the inlet surface of the MPP cell. The
observed same profiles demonstrate that the flow is fully
developed before reaching the MPP cell.
The corresponding semi-log plots of the mean stream-
wise velocity at x¼�50 mm is shown in Fig. 4, along with
the well-known logarithmic law.40 In the figure, Uþ ¼ hUi=Us, hUi is the mean stream-wise velocity. It can be seen that
the predicted data agree well with the empirical equation,40
including the viscous sublayer region (yþ � 5) where the
CFD data follow quite well with the well-known trend
Uþ ¼ yþ.
B. Friction velocity
As to be demonstrated later, the acoustic impedance of
the MPPs under grazing flow is well correlated to the
FIG. 2. System model.
FIG. 3. (Color online) Mean x-velocity profiles at different locations upstream
the resonator, through the y¼ 0 cross section. M¼ 0.25.
FIG. 4. (Color online) Semi-log plots of the mean stream-wise velocity profile
upstream the resonator at x¼�50 mm, through y¼ 0 cross section. M¼ 0.25.
J. Acoust. Soc. Am. 145 (4), April 2019 Xiaoqi Zhang and Li Cheng 2463
velocity gradient in the viscous sublayer over the duct wall,
G, defined as
G ¼ U2s
v; (1)
with the friction velocity, Us, calculated by
Us ¼ffiffiffiffiffisw
q
r; (2)
where q is the air density, sw is the wall shear stress, which
can be determined from the free-stream grazing flow veloc-
ity U and the Darcy friction factor as
sw ¼qU2k
8: (3)
The expression developed by Fujita41 is used to calculate kwhose accuracy is shown to be within engineering accuracy19
k ¼ 0:178
R1=5e
; (4)
where Re ¼ hU=v is the Reynolds number with h being the
height of the square duct.
The numerically calculated friction velocity is compared
with the prediction by the empirical equation,41 as shown in
Fig. 5. The agreement between the two sets of results dem-
onstrate the validity of the computational model.
C. Acoustic impedance simulation and comparisonswith experiments
The accuracy of the CFD model to predict the acoustic
impedance of MPPs in the presence of grazing flow is vali-
dated through comparisons with experimental data reported
in the literature.
The normalized acoustic impedance of a MPP hole is
defined as
Zhole ¼1
qc
Pin � Pout
�u; (5)
where c is the speed of sound in air, Pin and Pout are the
space-averaged acoustic pressure over the inlet and outlet
surface of the hole, respectively, and �u is the space-averaged
acoustic velocity normal to the hole cross-section. Since the
acoustic wavelength is much larger than the thickness of the
hole, the normal acoustic velocities at both sides of the hole
can be assumed to be the same.
The normalized acoustic impedance of the entire MPP
can then be obtained by
Z ¼ Zhole
d¼ Rþ jv; (6)
where R and v are the normalized acoustic resistance and
reactance, respectively.
Fast Fourier transform is performed on the stable cycles
of the related time signals, and their complex values at the
acoustic frequency are determined.39 Using Eqs. (5) and (6),
the normalized acoustic impedance of MPP is then deduced.
The acoustic behavior of Panel 1 under different flow
speeds is first investigated, for a Mach number varying from
0.04 to 0.25 and an acoustic excitation at 111 dB at 3150 Hz.
Note this frequency is chosen to enable comparisons with
the data provided in Ref. 38. The Mach numbers alongside
the corresponding Reynolds numbers of the computation
cases are listed in Table II.
The acoustic impedance of Panel 1 obtained from the
CFD simulations and from experimental measurements,38 is
compared in Fig. 6. Both sets of results show that the acous-
tic resistance increases with the Mach number, while the
reactance decreases. It can also be seen that the experimen-
tally observed trend and the magnitude of the impedance
FIG. 5. (Color online) Comparison of the friction velocity between CFD
and the empirical equation (Ref. 41).
TABLE II. Computational cases.
Mach number 0.04 0.08 0.1 0.15 0.16 0.2 0.25
Re 21 108 43 271 52 769 79 153 84 430 105 538 131 923
FIG. 6. (Color online) Comparison of the normalized acoustic impedance of
Panel 1 between CFD and experiments.
2464 J. Acoust. Soc. Am. 145 (4), April 2019 Xiaoqi Zhang and Li Cheng
terms are reasonably well reproduced by the CFD
simulations.
Note that in the no-flow case, an end correction term
appears in the reactance part in Eq. (6) to count for the effec-
tive mass flowing through the hole. This, however, is diffi-
cult to be precisely determined in the present case, which
apparently depends on the flow conditions. The end-
correction, however, is not necessary to the resistance part
with flow, since dissipation mainly takes place inside the
hole. It is also pertinent to note that previous works16,18,24
have shown that, with grazing flow, the acoustic resistance is
insensitive to frequency variation (also confirmed in the fol-
lowing analyses). Therefore, the comparison, though at only
one frequency, is deemed representative enough.
The above validations confirm the validity of the accept-
able accuracy of the proposed CFD model, which is to be
used for flow analyses and the development of the acoustic
impedance prediction formula.
IV. FLOW FIELD VISUALIZATIONS AND ANALYSES
A. Flow field at different Mach numbers
Analyses on the flow field near the MPP aperture and
inside the hole would help to better understand the physical
process of the flow-acoustic-MPP interaction. To this end,
the case without acoustic excitation is first examined. Figure
7 shows the velocity streamlines of the mean grazing flow
near the hole of the resonator at different Mach numbers. It
can be seen that the flow passes over the hole with no visible
fluid flowing into or out of the cavity through the hole, along
with the formation of a shear layer above it. The grazing
flow induces an additional flow motion inside the hole by the
shear stress transmitted through the hole in the wall. At a
low Mach number of M¼ 0.04, two vortical flow regions in
the hole are observable. In the upper region a big vortex with
clockwise rotation is formed, entraining a smaller one in the
lower area with counter clockwise rotation. The vortices
together with the small hole would prevent the fluid entrain-
ment through the hole. When the flow velocity increases, the
smaller vortex in the lower part of the hole disappears while
the upper vortical flow region grows, leading to an increase
in the contacting area between the vortical flow and the wall
of the hole.
Acoustic excitation is then added. As an example, the
velocity streamlines near the hole with an acoustic excitation
defined in Sec. III C are shown in Fig. 8. It can be seen that,
at a very low Mach number of M¼ 0.04, different from the
sole mean grazing flow case (Fig. 7), fluid is entrained into
the cavity. Meanwhile, the vortex in the upper area of the
hole becomes smaller and the small vortex in the lower area
of the hole with counter clockwise rotation disappears, as
compared with the case without acoustic excitation (Fig. 7).
However, the effect of the acoustic excitation on the global
flow field is not obvious at other flow velocities. This can be
explained by the fact that the size of the vortex in the hole
gradually grows with the flow speed, along with an increase
of the resistant effect of the vortex in the hole. In the present
case, the acoustic energy is not strong enough to overcome
the resistant effect of the vortex in the hole at high flow
velocities. As a result, unlike the case of M¼ 0.04, there is
no fluid flowing into the cavity, and the streamline pattern
near the hole is almost the same as the case without acoustic
excitation. The observed flow pattern also echoes with the
common belief that viscous dissipation in the shear layer of
the hole dominates the sound absorption mechanism at this
typical SPL level.
B. Intrinsic flow parameters for acoustic impedanceprediction
The flow field near the hole can be seen as a superposi-
tion of the oscillating flow field caused by the acoustic exci-
tation and the grazing flow. It can be surmised that the flow
parameter which can best determine the flow field near the
FIG. 7. Velocity streamlines of mean grazing flow near the hole of the reso-
nator through y¼ 0 cross section at different flow Mach numbers. (a)
M¼ 0.04, (b) M¼ 0.1, and (c) M¼ 0.25.
J. Acoust. Soc. Am. 145 (4), April 2019 Xiaoqi Zhang and Li Cheng 2465
hole would influence the acoustic impedance of the MPP
holes. An enlightening example is the case of a linear shear
flow passing over a plane wall with a circular hole, generat-
ing the so-called Stokes flow near the hole,42–46 in which
case the local Reynolds number based on the diameter of the
orifice is smaller than 1. The exact solution by Davis46 sug-
gests that the velocity gradient of the linear shear flow might
be the key parameter which determines the distribution of
the velocity and pressure near the hole. Enlightened by the
work of Davis,46 we propose the flow velocity gradient in
the viscous sublayer as the new flow parameter, and subse-
quently establish its link to the acoustic impedance of the
MPP hole. The choice of the velocity gradient is also sup-
ported by the flow field analyses, reported in Sec. IV A. With
the viscous sublayer being adjacent with the flow field in the
hole, the velocity gradient in this layer produces the shear
stress, responsible for the flow pattern in the hole as well as
the acoustic energy dissipation.
V. IMPEDANCE PREDICTION MODEL
A. Relationship between the velocity gradient and theacoustic resistance of MPPs
To establish the relationship between the velocity gradi-
ent in the viscous sublayer over duct wall, denoted by G, and
the acoustic resistance of MPPs under a grazing turbulent
flow, numerical experiments are curried out. Panel 1, the
same one used in Ref. 38, is first investigated. After finding
the relationship between the velocity gradient in the viscous
sublayer and the acoustic resistance of this panel, other pan-
els will be used for further verifications. For each panel,
additional computational cases are considered with details
tabulated in Table III. In each case, the panel is exposed to
the same flow speed range and acoustic excitation level as
before, but at different frequencies.
In the no-flow case, the normalized acoustic resistance
of a MPP proposed by Maa4 writes
R ¼ Rin þ Rout ¼32vt
dcd21þ K2
32
� �1=2
þffiffiffi2p
32K
d
t; (7)
where K ¼ dffiffiffiffiffiffiffiffiffiffiffix=4v
p; Rin represents the viscous effects in
the hole and Rout is the end correction counting for the vis-
cous loss outside the hole.
In the presence of grazing flow, we replace Rout with a
new term, h, to count for the viscous loss. Therefore, the nor-
malized acoustic resistance of the MPP, Rflow, writes
Rflow ¼ Rin þ h: (8)
Both h and G are non-dimensionalized as hc=fd and
Gt=fd, respectively. Figure 9 displays the CFD results using
Panel 1 at different excitation frequencies ranging from 2400
to 3300 Hz at different flow velocities. The plot is grouped
into four groups, each having the same Mach number but
different frequencies. The corresponding Mach number
varies from 0.1 to 0.25. As observed in Fig. 9, although each
group of results seems to have a slightly different slope,
there seems to exist a rather linear relationship between the
two parameters (hc=fd and Gt=fd) which can be reasonably
well represented by a straight line. This also confirms that
the amended term in the acoustic resistance formula under
grazing flow is related to the velocity gradient in the viscous
sublayer region.
To further confirm the generality of the observed linear
relationship between, two other MPPs (Panels 2 and 3) with
different hole dimensions are also examined through CFD
simulations (with cases listed in Table III). Results are
shown in Fig. 10, which show that, though following a dif-
ferent slope, the linear relationship between hc=fd and Gt=fdstill holds reasonably well, confirming the general nature of
the observations made on Panel 1.
A linear regression analysis is conducted to establish the
observed linear relationship as
hc
f d¼ A
Gt
f dþ B; (9)
FIG. 8. Velocity streamlines near the hole of the resonator through y¼ 0
cross section at different flow velocities. f ¼ 3.15 kHz, jVaj ¼ 0.025 m/s. (a)
M¼ 0.04, (b) M¼ 0.1, and (c) M¼ 0.25.
2466 J. Acoust. Soc. Am. 145 (4), April 2019 Xiaoqi Zhang and Li Cheng
where A and B, as a function of t/d, are to be determined.
Curve-fitting the calculated resistance data for panels 1, 2,
and 3 leads to the following expression:
hc
f d¼ 0:0356
t
d
� ��3:236
þ 0:0157
" #Gt
dfd
þ 1:369� 2:331t
d
� ��2:195" #
1
d: (10)
B. Resistance model of MPPs and validations
Combining Eqs. (8) and (10), a normalized acoustic
resistance formula for MPPs under grazing turbulent flow is
established as follows:
Rflow ¼ Rin þ 0:0356t
d
� ��3:236
þ 0:0157
" #Gt
dc
þ 1:369� 2:331t
d
� ��2:195" #
f d
dc; (11)
where Rin ¼ ð32vt=dcd2Þ½1þ ðK2=32Þ�1=2, with G calculated
by Eqs. (1)–(4).
For validation purposes, Fig. 11 shows a comparison of
the normalized acoustic resistance between the proposed for-
mula and the experimental data for Panel 1, reported in Ref.
38. Meanwhile, the same comparisons with the CFD data for
Panel 1 at other frequencies are also given in Fig. 12 with
different Mach numbers. These comparisons clearly show
that the proposed formula cannot only capture the trend but
also agrees well with both the experimental data and CFD
simulations. They also confirm the fact that the resistance is
nearly constant and insensitive to frequency variations.
Similar conclusions can be drawn for Panels 2 and 3 (not
shown here for brevity).
It should also be pointed out that the above analyses and
the proposed acoustic resistance formula are based on a
number of selected cases. Therefore, the validity range, as
verified in the present study, is limited to Re� 131 923 (cor-
responding to M � 0:25 for current cases), 1 � ðt=dÞ � 2
and under linear acoustic excitation range.
Additional comparisons with other existing models
reported in the literature are finally conducted. To this end,
Kirby and Cummings’ model19 and Allam and Abom’s
model24 are used, with results shown in Fig. 13. To facilitate
comparisons, experimental data38 are also provided in the
same figure. It can be seen that the prediction results by the
present formula seem to fit the measured data better than the
other two models. More specifically, compared with Kirby
and Cummings’ model, the present model seems to work
better, especially at a low Mach number range before 0.1. In
TABLE III. Computational cases used to find the relation between G and the
acoustic resistance.
Velocity amplitude
of the incident
acoustic wave (m/s) Frequency (Hz) Mach number Re
jVaj ¼ 0.025 3150, 3000, 2800, 2600, 2400 0.1 52 769
3300, 3150, 3000, 2800, 2600, 0.15 79 153
3300, 3150, 3000, 2800, 2600, 0.2 105 538
3300, 3150, 3000, 2800, 2600, 0.25 131 923
FIG. 9. (Color online) Relationship between h and G for Panel 1. jVaj¼ 0.025 m/s.
FIG. 10. (Color online) Relationship between h and G for panels 2 and 3.
jVaj ¼ 0.025 m/s.
FIG. 11. (Color online) Normalized acoustic resistance comparisons between
proposed formula and experimental data for panel 1. f ¼ 3.15 kHz, jVaj¼ 0.025 m/s.
J. Acoust. Soc. Am. 145 (4), April 2019 Xiaoqi Zhang and Li Cheng 2467
the higher Mach number region, the proposed formula fol-
lows better experimental data than Allam and Abom’s
model. Over the entire region, the proposed model can cap-
ture the nonlinearly increasing trend of the acoustic
resistance.
As mentioned before, both previous work16,18,24 and the
CFD simulations in Fig. 12 show that, with grazing flow, the
acoustic resistance is roughly independent of frequency.
Therefore, the experimental data found in literature, though
at one frequency, is rather representative. Similarly, the lim-
ited cases being investigated in the paper actually cover a
much wider scope and are applicable for a range of different
frequencies. However, considering the very scarce experi-
mental research on MPPs with low grazing flow velocity
under the linear acoustic excitation region, more experiments
will be carried out in our future work.
As a final remark, it should be noted that the above anal-
yses heavily focus on the acoustic resistance part of the
impedance formula. To complete the acoustic impedance
prediction, the proposed resistance formula can be combined
with the acoustic reactance prediction model proposed by
Cummings,26 which was tested to show a rather good consis-
tency against the experimental results on Panel 1 (results not
shown here for brevity). Casting Cummings’ reactance for-
mula by using parameter G proposed in this paper is straight-
forward. Combined with Eq. (11), a complete set of the
acoustic impedance prediction formulas for MPPs with graz-
ing flow within the linear acoustic regime is established.
VI. CONCLUSIONS
The acoustic behavior of MPPs, exposed to fully devel-
oped grazing turbulent flow, is investigated through numeri-
cal simulations. Grazing flow and acoustic wave-excited
flow field near the orifice of the MPP under different flow
conditions is scrutinized through solving the compressible
Reynolds averaged Navier–Stokes equations.
Based on the analyses of the flow field near the MPP ori-
fice, the velocity gradient in the viscous sublayer over the
duct wall is identified as the intrinsic flow parameter which
is inherently related to the acoustic resistance of the MPPs
with grazing flow. Numerical results reveal its linear rela-
tionship with the flow-related part in the acoustic resistance
formula. Based on this, a new resistance formula is pro-
posed, which is shown to be applicable at a Mach number up
to roughly 0.25 under the linear acoustic excitation regime.
The accuracy and superiority of the model as compared with
the existing ones are demonstrated through comprehensive
comparisons with the data provided in the open literature. It
is shown that the proposed formula agrees well with the
experimental data and outperforms existing models in terms
of both prediction accuracy and application range.
Combined with Cummings’ reactance model, a complete set
of the acoustic impedance prediction formula for MPPs with
grazing flow is established.
ACKNOWLEDGMENTS
The authors thank the support from Research Grant
Council of the Hong Kong SAR (PolyU 152036/18 E).
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